 Award ID(s):
 2106906
 NSFPAR ID:
 10334996
 Date Published:
 Journal Name:
 Journal of Topology and Analysis
 ISSN:
 17935253
 Page Range / eLocation ID:
 1 to 45
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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We consider two manifestations of nonpositive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, rightangled Coxeter groups, most 3–manifold groups, rightangled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that rightangled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasi convexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and rightangled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known.more » « less

We consider two manifestations of nonpositive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, rightangled Coxeter groups, most 3 3 –manifold groups, rightangled Artin groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a “best” one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that rightangled Coxeter groups admit universal acylindrical actions. The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and rightangled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known. In the appendix, it is verified that any space satisfying the a priori weaker property of being an “almost hierarchically hyperbolic space” is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.more » « less

Abstract Actions on hyperbolic metric spaces are an important tool for studying groups, and so it is natural, but difficult, to attempt to classify all such actions of a fixed group. In this paper, we build strong connections between hyperbolic geometry and commutative algebra in order to classify the cobounded hyperbolic actions of numerous metabelian groups up to a coarse equivalence. In particular, we turn this classification problem into the problems of classifying ideals in the completions of certain rings and calculating invariant subspaces of matrices. We use this framework to classify the cobounded hyperbolic actions of many abelian‐by‐cyclic groups associated to expanding integer matrices. Each such action is equivalent to an action on a tree or on a Heintze group (a classically studied class of negatively curved Lie groups). Our investigations incorporate number systems, factorization in formal power series rings, completions, and valuations.

We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups.more » « less

Properly discontinuous actions of a surface group by affine automorphisms of ℝ^d were shown to exist by DancigerGueritaudKassel. We show, however, that if the linear part of an affine surface group action is in the Hitchin component, then the action fails to be properly discontinuous. The key case is that of linear part in 𝖲𝖮(n,n−1), so that the affine action is by isometries of a flat pseudoRiemannian metric on ℝ^d of signature (n,n−1). Here, the translational part determines a deformation of the linear part into 𝖯𝖲𝖮(n,n)Hitchin representations and the crucial step is to show that such representations are not Anosov in 𝖯𝖲𝖫(2n,ℝ) with respect to the stabilizer of an nplane. We also prove a negative curvature analogue of the main result, that the action of a surface group on the pseudoRiemannian hyperbolic space of signature (n,n−1) by a 𝖯𝖲𝖮(n,n)Hitchin representation fails to be properly discontinuous.more » « less